The sixth-order Krall differential expression and self-adjoint operators.

Abstract

We first provide an overview of classical GNK Theory for symmetric, or symmatrizable, differential expressions in L2((a, b);w). Then we review how this theory was applied to find a self-adjoint operator in L2 \mu(-1, 1) generated by the sixth-order Lagrangian symmetric Krall differential equation, as done by S. M. Loveland. We later construct the self-adjoint operator generated by the Krall differential equation in the extended Hilbert space L2(-1, 1) + C2 which has the Krall polynomials as (orthogonal) eigenfunctions. The theory we use to create this self-adjoint operator was developed recently by L. L. Littlejohn and R. Wellman as an application of the general Glazman-Krein-Naimark (GKN) Theorem discovered by W. N. Everitt and L. Markus using complex symplectic geometry. In order to explicitly construct this self-adjoint operator, we use properties of functions in the maximal domain in L2(-1, 1) of the Krall expression. As we will see, continuity, as a boundary condition, is forced by our construction of this self-adjoint operator. We also construct six additional examples of self-adjoint operators in an extended Hilbert space, three with a one-dimensional extension space and three with a two-dimensional extension space.